To solve this problem, let’s go through each part step by step.

Step 1: Find the Mean of the Sample Data

The temperatures measured are: $77.5, 74.9, 67.3, 83.9, 72.2, 86.7, 80.4, 71, 72.9, 86.7, 73.6, 86.8$

Let's calculate the sample mean $\bar{x}$: $\bar{x} = \frac{77.5 + 74.9 + 67.3 + 83.9 + 72.2 + 86.7 + 80.4 + 71 + 72.9 + 86.7 + 73.6 + 86.8}{12}$

Step 2: Construct a 99% Confidence Interval for the Mean

1. The sample size $n = 12$.
2. The population standard deviation $\sigma = 6.88$.
3. For a 99% confidence level, the z-score (critical value) $z_{\alpha/2}$ is approximately 2.576.

The formula for a confidence interval is: $\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$

Step 3: Calculate the 95% Single-Sided Lower Bound

For a single-sided lower bound with 95% confidence, we use a z-score of approximately 1.645.

The lower bound is: $\bar{x} - z \frac{\sigma}{\sqrt{n}}$

Step 4: Determine the Sample Size Needed for a Desired Margin of Error

We need to find the required sample size $n$ to have a margin of error $E < 1$ with 95% confidence. The formula for sample size with a given margin of error is: $n = \left( \frac{z \sigma}{E} \right)^2$ where $E = 1$ and $z = 1.645$.

Let me calculate each of these steps.Here are the results for each part:

1. Sample Mean ($\bar{x}$): 77.825

2. 99% Confidence Interval for the Mean: $72.71 \leq \mu \leq 82.94$

3. 95% Single-Sided Lower Bound for the Mean: $\mu \geq 74.56$

4. Required Sample Size for Margin of Error $E < 1$ with 95% Confidence: $n = 129$

If you need more details or have questions, let me know!

Related Questions

1. What does a 99% confidence interval indicate about the mean?
2. How is the margin of error affected by the sample size?
3. Why is a larger sample size required for a smaller margin of error?
4. How would the confidence interval change if the standard deviation were different?
5. What if we wanted a 90% confidence interval—how would the z-score change?

Tip

Remember, confidence intervals provide a range within which we expect the true population parameter to lie, based on the sample data and a specified confidence level.